Classification of extremal and $s$-extremal binary self-dual codes of length 38

TL;DR

This paper classifies all extremal and s-extremal binary self-dual codes of length 38, identifying their quantities and types using a recursive algorithm and its generalization, advancing the understanding of these codes.

Contribution

It provides the first complete classification of extremal and s-extremal binary self-dual codes of length 38, utilizing a novel recursive algorithm and its extension.

Findings

2744 extremal [38,19,8] codes identified

Two s-extremal [38,19,6] codes found

1730 s-extremal [38,19,8] codes classified

Abstract

In this paper we classify all extremal and $s$-extremal binary self-dual codes of length 38. There are exactly 2744 extremal $[38,19,8]$ self-dual codes, two $s$-extremal $[38,19,6]$ codes, and 1730 $s$-extremal $[38,19,8]$ codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of $s$-extremal $[38,19,6]$ codes permits to achieve the classification of all $s$-extremal codes with d=6.